Download Topic 6 Real and Complex Numbers II

Topic 6 Real and Complex Number Systems II
9.1 – 9.5, 12.1 – 12.2
Algebraic representation of complex numbers including:
• Cartesian, trigonometric (mod-arg) and polar form
• definition of complex numbers including standard and
trigonometric form
• geometric representation of complex numbers including
Argand diagrams
• powers of complex numbers
• operations with complex numbers including addition,
subtraction, scalar multiplication, multiplication and
conjugation
Topic 6
Real and Complex Number
Systems II
2
i
Definition
= -1
 i = -1
A complex number has the form
z = a + bi
(standard form)
where a and b are real numbers
We say that Re(z) = a
and that Im(z) = b
[the real part of z]
[the imaginary part of z]
i =i
i2 = -1
i3 = -i
i4 = 1
i5 = i
i6 = -1
i7 = -i
i8 = 1
Question : What is the value of i2003 ?
Model : Solve x  2 x  2  0
2
 b  b  4ac
x
2a
2
 2  2  4 21

2
2 4

2
 2  2i

2
 1  i
2
Equality If a + bi = c + di
then a = c and b = d
Addition a+bi + c+di = (a+c) + (b+d)i
e.g. 3+4i + 2+6i = 5+10i
e.g. 2+6i – (4-5i) = 2+6i-4+5i
= -2+11i
Scalar Multiplication 3(4+2i) = 12+6i
Multiplication (3+4i)(2+5i)
= 6+8i+15i+20i2
= 6 + 23i + -20
= -14 + 23i
(2+3i)(4-5i)
= 8-10i+12i-15i2
= 8 + 2i -15 i2
= 23 + 2i
In general (a+bi)(c+di) = (ac-bd) + (ad+bc)i
Exercise
Exercise
NewQ P 227, 234
Exercise 9.1, 9.3
FM P 168
Exercise 12.1
Determine the nature of the roots of each of the
following quadratics:
(a) x2 – 6x + 9 = 0
(b) x2 + 7x + 6 = 0
(c) x2 + 4x + 2 = 0
(d) x2 + 4x + 8 = 0
Determine the nature of the roots of each of the following quadratics:
(a) x2 – 6x + 9 = 0
(b) x2 + 7x + 6 = 0
(c) x2 + 4x + 2 = 0
(d) x2 + 4x + 8 = 0
(a) x2 – 6x + 9 = 0
= 36 – 4x1x9
=0
∴ The roots are real and equal
[x=3]
Determine the nature of the roots of each of the following quadratics:
(a) x2 – 6x + 9 = 0
(b) x2 + 7x + 6 = 0
(c) x2 + 4x + 2 = 0
(d) x2 + 4x + 8 = 0
(b) x2 + 7x + 6 = 0
= 49 – 4x1x6
= 25
∴ The roots are real and unequal
[ x = -1 or -6 ]
Determine the nature of the roots of each of the following quadratics:
(a) x2 – 6x + 9 = 0
(b) x2 + 7x + 6 = 0
(c) x2 + 4x + 2 = 0
(d) x2 + 4x + 8 = 0
(c) x2 + 4x + 2 = 0
= 16 – 4x1x2
= 8
∴ The roots are real, unequal and irrational
[ x = -2  2 ]
Determine the nature of the roots of each of the following quadratics:
(a) x2 – 6x + 9 = 0
(b) x2 + 7x + 6 = 0
(c) x2 + 4x + 2 = 0
(d) x2 + 4x + 8 = 0
(d) x2 + 4x + 8 = 0
= 16 – 4x1x8
= -16
∴ The roots are complex and unequal [ x = -2  4i ]
Exercise
FM P 232
Exercise 9.2
Division of complex numbers
3  2i
Model
4  3i
3  2i 4  3i


4  3i 4  3i
12  9i  8i  6i 2

16  9
6  17 i

25
6 17


i
25 25
Try this on your GC
Exercise
NewQ P 239
Exercise 9.4
Exercise
• Prove that the set of complex numbers
under addition forms a group
• Prove that the set of complex numbers
under multiplication forms a group
Model : Show that the set {1,-1,i,-i} forms a
group under multiplication
x
1
-1
i
-i
1
1
-1
i
-i
-1
-1
1
-i
i
i
i
-i
-1
1
-i
-i
i
1
-1
• Since every row and column contains every element , it
must be a group
Exercise
NewQ P 245
Exercise 9.5
Argand Diagrams
Model : Represent the complex number
3+2i on an Argand diagram
or
Model : Show the addition of 4+i and 1+2i on an Argand
diagram
4
y
2
x
0
-6
-4
-2
0
-2
-4
2
4
6
Draw the 2 lines representing these numbers
4
y
2
x
0
-6
-4
-2
0
-2
-4
2
4
6
Complete the parallelogram and draw in the diagonal.
This is the line representing the sum of the two numbers
4
y
2
x
0
-6
-4
-2
0
-2
-4
2
4
6
Exercise
New Q P300
Ex 12.1
Model : Express z=8+2i in mod-arg form
y
4
(8,2)
2
x
0
-10
-8
-6
-4
-2
0
-2
-4
2
4
6
8
10
Model : Express z=8+2i in mod-arg form
x
r
x  r cos 
y
r
y  r sin 
cos  
sin  
8  2i  x  iy
 r cos   r i sin 
 r cos   i sin  
 r cis
y
4
(8,2)
2
r

0
-10
-8
-6
-4
-2
0
-2
-4
y
2
x
4
x
6
8
10
Model : Express z=8+2i in mod-arg form
r

82  22
68
tan   82
  14o
y
z  8  2i 
68 cis 14o
4
(8,2)
2
r

0
-10
-8
-6
-4
-2
0
-2
-4
2
x
4
6
8
10
Model : Express
4

3i
4
in mod  arg form
3i
4
3i

3i
3i
4( 3  i )

31

y
3  i  r cis
3i
3
r  ( 3 ) 2  12
2
2
tan  
 
1
1
3
r

6
 r cis  2 cis 6
0
-1
0
θ
x
1
2
3
Model: Express 3 cis /3 in standard form
3 cis 3
 3(cos 3  i sin 3 )


 3(  i
3
2
1
2
 
3
2
3 3
2
i
)
Exercise
New Q P306
Ex 12.2